Can't remove an unselectable face
The reason you cant remove that face individually is because it's part of another face. Your object has at least one concave n-gon (a face with more than 4 sides where one of the sides comes back towards the center of the face) and that is what is causing you your problem. All the faces of an object are converted into triangles before they can be displayed, and with the faces on your object it appears the triangulation process causes a face to go outside the boundary of the n-gon. You can better visualize what I mean if you select the face (the one shown in the bottom screenshot) and press Ctrl T to convert it to triangles. You will then see one of the triangles causing the issue.

The best way to solve the issue is not to delete the face, but instead help blender out by giving it smaller, convex faces to work with. I can't see your entire mesh in the screenshots, but you should be able to select one of the vertices near where the issue is, then select another vertex somewhere on another side of the face either below or to the right. If you press J then it will connect those two vertices into an edge and split the large face into two smaller ones that should be easier for blender to triangulate.

In general it's best to avoid faces with more than 4 sides. The only place n-gons are usually ok is when they are perfectly flat as well as convex (think classic pentagon or similar). If you can, it would be best to convert you mesh to being made out of 4 sided faces only. It will be easier for blender to display without any glitches, it will save you headaches down the road if you have to work with this object again, and it will also be good practice to be able to make any object with good geometry.